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Some Properties of an Archimedean $\Ell $-Group Czechoslovak Mathematical Journal Dao Rong Tong Some properties of an archimedean `-group Czechoslovak Mathematical Journal, Vol. 45 (1995), No. 2, 293–302 Persistent URL: http://dml.cz/dmlcz/128525 Terms of use: © Institute of Mathematics AS CR, 1995 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz Czechoslovak Mathematical Journal, 45 (120) 1995, Pгaha SOME PROPERTIES OF AN ARCHIMEDEAN £-GROUP DAO-RONG TON, Nanjing (Received March 30, 1993) 1. AUXILIARY RESULTS We will use the standard notation for ^-groups, cf. [5]. Throughout the paper G is an ^-group, R is the real group, Q is the rational group and Z is the integer group. If G and H are ^-groups, G ffl H denotes their cardinal sum. Let {Ga \ a G A} be a we system of ^-groups and let Yl Ga be their product. For an element g G Yl Ga aeA aeA denote the a component of g by ga. An £-group G is said to be a subdirect sum of ^-groups GQ, in symbols G C' y[ GQ, if G is an ^-subgroup of Yl Ga such that for aeA aeA each a G A and each g' G Ga there exists g G G with the property gQ = g'. An £- group G is said to be an ideal subdirect sum of ^-groups Ga, in symbols G C* Yl Ga, aeA if G C' Yl Ga and G is an £-ideal of Yl Ga. We denote the ^-subgroup of Yl Ga aeA aeA aeA consisting of the elements with only finitely many non-zero components by ^ Ga. aeA An f-group G is said to be a completely subdirect sum, if G is an ^-subgroup of J! Ga and £ GQ C G. aeA aeA A subset {0} ^ D C G is said to be disjoint, if Oi A g2 = 0 for any pair of distinct elements gi, g-2 € D. For any A" C G we designate X1- = {g G G | \g\ A |x| = 0 for each T G A"}. For g G G, [O] is the convex ^-subgroup of G generated by O, (g) = g11 is the polar subgroup of G generated by g. We denote the least cardinal a such that |A| ^ a for each bounded disjoint subset A of G by UG, where |A| denotes the cardinal of A. G is said to be ^-homogeneous if vH = vG for any convex ^-subgroup II ^ {0} of G. If G is an archimedean ^-homogeneous £-group and vG = K;, we call G an archimedean D-homogeneous t^-group of K; type. In [9] we proved that an ^-group G is complete if and only if G is ^-isomorphic to an ideal subdirect sum of real groups, integer groups and continuous U-homogeneous complete ^-groups. By using this result, we described the structure of an archimedean 293 £-group in [10]. Suppose that G is a subdirect sum of subgroups of reals and v- homogeneous ^-groups, G C' Yl Ts. Let Ai = {l5 G A | T^ is a subgroup of reals}. SEA If ~l Ts C G, then G is said to be a semicomplete subdirect sum of subgroups of 6eAl reals and ^-homogeneous ^-groups of Hi type, in symbols (1.1) £ T6CGC' l\Ts. 8eAxCA 8<EA Theorem 1.1 (Theorem 4.7 of [10]). An E-group G is archimedean if and only if G is ^-isomorphic to a semicomplete subdirect sum of subgroups of reals and archimedean v-homogeneous i-groups of Hi type. Now let G be an archimedean ^-group. Then we have an ^-isomorphism Q such that £ TSl C QG C' TJ TS, (5i€AiCA SeA where T$x is a subgroup of reals for each 6\ G Ai C A and Ts is an archimedean v- homogeneous ^-group of Hi type for each 6 G A \ Ai. For x G G put xl — (... x\ .. .) such that ! _ j(Qx)s ^G Ai, X& ~ jo (56 A\Ai. We call x1 the real part of x. If for any x G G, the real part T1 G OG, G is said to be real decomposable archimedean £-group. In this case, if we put x2 = (... x2 ...) as follows: '0 «GAi, Xs 1 (QX)S SeA\Al then 1 2 QX — x -f x , 1 and Æ2 — QX - x Є QG. Put Gi = {Qxe QG \X EG,(QX)S =0for 6 e A\ Ai}, G2 = {DxG DG | x G G, (DT)* =0for(5G Ai}. Then both Gi and G2 are ^-subgroups of DG, moreover, DG = Gi fflG2. ± It is clear that G2 = R(QG) (the radical of G) and Gi = R(QG) . 294 Corollary 1.2. Let G be a real decomposable archimedean (-group. Then G is (^-isomorphic to a cardinal sum G\ EB G2, where G\ is a completely subdirect sum of subgroups of reals and G2 is a subdirect sum of archimedean v-homogeneous £-groups of^i type. So, if G is a real decomposable archimedean £-group, then G = R(G) ffl H(G)±. However, in general, R(G) is not a cardinal summand of G. If G is complete or laterally complete, then R(G) is a cardinal summand. 2. A COMPLETELY SUBDIRECT SUM OF SUBGROUPS OF REALS Now we can characterize those ^-groups which can be represented as completely subdirect sums of subgroups of reals. Theorem 2.1. Let G 7-= {0} be an archimedean (.-group. Then the following properties are equivalent: (1) G is (-isomorphic to a completely subdirect sum of subgroups of reals; (2) G is (-isomorphic to an ideal subdirect sum of real groups and integer groups; (3) G has a basis. Proof. (1) => (2): Without loss of generality, assume 6eA seA where each Ts is a subgroup of R for S G A. Then A G C* ft Ts\ SeA where Tf = R or Z for S € A. (2) -=> (1): It is similar to the proof of Theorem l.L (1) =-> (3): If we have the formula (1.1), then for each S (E A we choose a fixed ts with 0 < ts G Ts; the system {ts \ S € A} is a basis for G. (3) => (2): See Theorem 3.5 in [5]. • By Theorem 4 and Corollary IV of Chapter 3 in [5] we see that an archimedean £-group G has a finite basis if and only if G is ^-isomorphic to a completely subdirect sum of a finite number of subgroups of reals. However, a completely subdirect sum of a finite number of subgroups of reals is a cardinal sum of a finite number of subgroups of reals. So we get 295 Corollary 2.2. An archimedean (-group G has a finite basis if and only if G is (.-isomorphic to a cardinal sum of a finite number of subgroups of reals. 3. HYPER-ARCHIMEDEAN PROPERTY An £-group G is called hyper-archimedean if each £-homomorphic image of G is archimedean. Proposition 3.1. An (-group G is hyper-archimedean if and only if G is pro­ tectable and [g] = (g) for each 0 < g G G. Proof. Necessity. Suppose that G is hyper-archimedean. For any 0 < g G G we have gL ffl (g) C G. But gL ffl (g) D gx ffl [g] = G by Theorem 2.4 in [5]. So G = g1- ffl (g), and G is projectable. From G = gx ffl [g] = gx ffl (g) we get [g] = (g). Sufficiency. If G is projectable and 0 < g G G, then G = gx ffl (g). Since [g] = (g), G = gx ffl [g]. Hence G is hyper-archimedean. • An £-group G is an a-extension of an ^-group H if and only if H is an ^-subgroup of G and the map L —)• LDH is a one-to-one map of the set of all convex ^-subgroups of G onto those of H. G is a-closed if it admits no proper a-extension. Corollary 3.2. Let G be a hyper-archimedean (-group with a basis. If G is a-closed, then G/P ~ R for each proper prime P. Proof. Let G be an a-closed hyper-archimedean l^-group with a basis. By the above Theorem 2.1, without loss of generality, we have ]TT, CGC' J]T,, seA seA where each Ts is a subgroup of reals. Let P be a proper prime. By Theorem 2.4 in [5] P is maximal and P = {x G G | xs{) = 0 for some S0 G A}. So G = TSll ffl P and G/P ~ Ts0. If G/P0 fails to be isomorphic to R for some proper prime P0, then G' = R ffl Fo 2 Q 3 -°o or G' = H ffl P0 2 Z ffl P0 is clearly an a-extension of G, a contradiction. Therefore G/P = H for each proper prime P. D 29G This corollary partly answers the question of the Corollary 2 in [2]. Next we discuss the hyper-archimedean kernel Ar(G) of an archimedean ^-group G. Ar(G) is a convex ^-subgroup of G which is hyper-archimedean and contains every hyper-archimedean convex ^-subgroup of G. An £-group G is continuous if for each 0 < x G G there exist x\, x2 G G such that x = x\ + x2, x\ A x2 = 0, x\ ^ 0 and #2 ^ 0. Lemma 3.3. Let G be a complete (laterally complete and archimedean) divisible v-homogeneous £-group of Hi type.
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